Today, in this otherwise simple how-to, we'll try to create a comprehensive list of of manufacturers and how they go about taking a screenshot. You'd think that it would be uniform throughout, but it actually isn't necessarily so, especially throughout the different platforms.
All iPhones behave uniformly in this regard. Just simultaneously press and hold the circular Home button and the Power/Lock button for a second and release. Voilà! The screenshot should be available in your Photos app, in the Camera Roll album.
Windows Phone 8- and Windows Phone 8.1-based devices use a different combination to take a screenshot.
For WP8, just press and hold the Start (Windows Logo) capacitive button and the Power/Lock button at the same time. If you have a WP8.1 device, then just press the Power/Lock button and Volume Up at the same time. Resulting screenshot will be available in your Photos Hub app, in a dedicated Screenshots album.
Some Android devices have a custom 'Take a Screenshot' button in their Power menu (see image on the right), available by holding down the Power button for two or so seconds. Don't worry, it'll take a screenshot of the content behind the menu itself.
If you have a Samsung phone, then you need to press down the Power/Lock button and the Home button simultaneously. Alternatively, certain Samsung phones allow you to take a screenshot by horizontally swiping the side of your hand across the entire display, like shown on your right.
If your Android version is not as up-to-date, then, sadly, you're unable to take screenshots as easily (unless you have a Samsung, where you can usually take a screenie with the Power/Lock button and Home button combo). If your device is rooted, however, quite a few free apps on the Play Store will allow you to snap your screen, like Screenshot Easy.